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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMay 27th 2022
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Idea An alternative to [[complete topological vector spaces]] in the framework of [[condensed mathematics]]. Roughly, completeness is expressed as ability to integrate with respect to [[Radon measures]]. This doesn't quite work as stated, and to make this rigorous one has to bring [[L^p-spaces]] for $0\lt p\le 1$ (i.e., the non-convex case) into the picture. ## Definition A [[condensed abelian group]] $V$ is __$p$-liquid__ ($0\lt p\le 1$) if for every [[compact Hausdorff topological space]] $S$ and every morphism of [[condensed sets]] $f\colon S\to V$ there is a unique morphism of [[condensed abelian groups]] $M_{<p}(S)\to V$ that extends $f$ along the inclusion $S\to M_{<p}(S)$. Here for a [[compact Hausdorff topological space]] $S$ and for any $p$ such that $0<p\le 1$ we have $$M_{<p}(S)=\bigcup_{q<p}M_q(S),$$ where $$M_p(S)=\bigcup_{C>0}M(S)_{\ell^p\le C},$$ where $$M(S)_{\ell^p\le C}=\lim_i M(S_i)_{\ell^p\le C},$$ where $S_i$ are finite sets such that $$S = \lim_i S_i$$ and $$M(F)_{\ell^p\le C}$$ for a [[finite set]] $F$ denotes the subset of $\mathbf{R}^F$ consisting of sequence with [[l^p-norm]] at most $C$. ## References See [[condensed mathematics]]. <a href="https://ncatlab.org/nlab/revision/liquid+vector+space/1">v1</a>, <a href="https://ncatlab.org/nlab/show/liquid+vector+space">current</a>

   Created:

   Idea

   An alternative to complete topological vector spaces in the framework of condensed mathematics.

   Roughly, completeness is expressed as ability to integrate with respect to Radon measures.

   This doesn’t quite work as stated, and to make this rigorous one has to bring L^p-spaces for $0\lt p\le 1$ (i.e., the non-convex case) into the picture.

   Definition

   A condensed abelian group $V$ is $p$-liquid ($0\lt p\le 1$) if for every compact Hausdorff topological space $S$ and every morphism of condensed sets $f\colon S\to V$ there is a unique morphism of condensed abelian groups $M_{<p}(S)\to V$ that extends $f$ along the inclusion $S\to M_{<p}(S)$.

   Here for a compact Hausdorff topological space $S$ and for any $p$ such that $0<p\le 1$ we have

   $$M_{<p}(S)=\bigcup_{q<p}M_q(S),$$

   where

   $$M_p(S)=\bigcup_{C>0}M(S)_{\ell^p\le C},$$

   where

   $$M(S)_{\ell^p\le C}=\lim_i M(S_i)_{\ell^p\le C},$$

   where $S_i$ are finite sets such that

   $$S = \lim_i S_i$$

   and

   $$M(F)_{\ell^p\le C}$$

   for a finite set $F$ denotes the subset of $\mathbf{R}^F$ consisting of sequence with l^p-norm at most $C$.

   References

   See condensed mathematics.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 28th 2022
   • (edited May 28th 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexhave fixed the link to [[p-norm|$\ell^p$-norm]]: [[p-norm|$\ell^p$-norm]] <a href="https://ncatlab.org/nlab/revision/diff/liquid+vector+space/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/liquid+vector+space/3">v3</a>, <a href="https://ncatlab.org/nlab/show/liquid+vector+space">current</a>

   have fixed the link to $\ell^p$-norm:

     [[p-norm|$\ell^p$-norm]]
   

   diff, v3, current